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Question

A class has three teachers, Mr. P, Ms. Q and Mrs.R and six students A,B,C,D,E,F. Number of ways in which they can be seated in a line of 9 chairs, if between any two teachers there are exactly two students, is k!(18), then the value of k is.

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Solution

Let T and S denotes teacher and student respectively.
Then we have following possible patterns according to question
(i) T S S T S S T S S
(ii) S T S S T S S T S
(iii) S S T S S T S S T
Hence total number of arrangements are 3(3!)6!=18×6!k=6.

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